Abstract

A simple Fourier transformation system working with broadband parallel illumination is presented. The proposed setup, consisting of two on-axis zone plates and an achromatic objective, allows us to obtain the achromatic Fourier transform representation of the input at a finite distance with a low chromatic aberration. The discussion of the system, using the Fresnel diffraction theory, leads to an analytical expression to evaluate the transversal and longitudinal chromatic aberrations. It is shown that the resulting chromatic aberrations for typical values of the involved parameters are less than 1% over the entire visible spectrum.

© 1992 Optical Society of America

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References

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  1. H. Bartelt, S. K. Case, R. Hauck, “Incoherent optical processing,” in Applications of Optical Fourier Transforms, H. Stark, ed. (Academic, New York, 1982).
  2. W. T. Rhodes, A. A. Sawchuk, “Incoherent optical processing,” in Optical Information Processing, S. H. Lee, ed. (Springer-Verlag, Berlin, 1981).
    [CrossRef]
  3. G. M. Morris, D. A. Zweig, “White-light Fourier transformations,” in Optical Signal Processing, J. L. Horner, ed. (Academic, New York, 1987).
  4. C. Brophy, “Design of an all-glass achromatic Fourier transform lens,” Opt. Commun. 47, 364–368 (1983).
    [CrossRef]
  5. G. M. Morris, “An ideal achromatic Fourier processor,” Opt. Commun. 39, 143–147 (1981).
    [CrossRef]
  6. G. M. Morris, “Diffraction theory for an achromatic Fourier transformation,” Appl. Opt. 20, 2017–2025 (1981).
    [CrossRef] [PubMed]
  7. R. Ferriere, J. P. Goedgebuer, “A spatially coherent achromatic Fourier transformer,” Opt. Commun. 42, 223–225 (1982).
    [CrossRef]
  8. R. Ferriere, J. P. Goedgebuer, “Achromatic systems for far-field diffraction with broadband illumination,” Appl. Opt. 22, 1540–1545 (1983).
    [CrossRef] [PubMed]
  9. S. Leon, E. N. Leith, “Optical processing and holography with polychromatic point source illumination,” Appl. Opt. 24, 3638–3642 (1985).
    [CrossRef] [PubMed]
  10. R. H. Katyl, “Compensating optical systems. Part 3: achromatic Fourier transformation,” Appl. Opt. 11, 1255–1260 (1972).
    [CrossRef] [PubMed]

1985 (1)

1983 (2)

1982 (1)

R. Ferriere, J. P. Goedgebuer, “A spatially coherent achromatic Fourier transformer,” Opt. Commun. 42, 223–225 (1982).
[CrossRef]

1981 (2)

1972 (1)

Bartelt, H.

H. Bartelt, S. K. Case, R. Hauck, “Incoherent optical processing,” in Applications of Optical Fourier Transforms, H. Stark, ed. (Academic, New York, 1982).

Brophy, C.

C. Brophy, “Design of an all-glass achromatic Fourier transform lens,” Opt. Commun. 47, 364–368 (1983).
[CrossRef]

Case, S. K.

H. Bartelt, S. K. Case, R. Hauck, “Incoherent optical processing,” in Applications of Optical Fourier Transforms, H. Stark, ed. (Academic, New York, 1982).

Ferriere, R.

R. Ferriere, J. P. Goedgebuer, “Achromatic systems for far-field diffraction with broadband illumination,” Appl. Opt. 22, 1540–1545 (1983).
[CrossRef] [PubMed]

R. Ferriere, J. P. Goedgebuer, “A spatially coherent achromatic Fourier transformer,” Opt. Commun. 42, 223–225 (1982).
[CrossRef]

Goedgebuer, J. P.

R. Ferriere, J. P. Goedgebuer, “Achromatic systems for far-field diffraction with broadband illumination,” Appl. Opt. 22, 1540–1545 (1983).
[CrossRef] [PubMed]

R. Ferriere, J. P. Goedgebuer, “A spatially coherent achromatic Fourier transformer,” Opt. Commun. 42, 223–225 (1982).
[CrossRef]

Hauck, R.

H. Bartelt, S. K. Case, R. Hauck, “Incoherent optical processing,” in Applications of Optical Fourier Transforms, H. Stark, ed. (Academic, New York, 1982).

Katyl, R. H.

Leith, E. N.

Leon, S.

Morris, G. M.

G. M. Morris, “Diffraction theory for an achromatic Fourier transformation,” Appl. Opt. 20, 2017–2025 (1981).
[CrossRef] [PubMed]

G. M. Morris, “An ideal achromatic Fourier processor,” Opt. Commun. 39, 143–147 (1981).
[CrossRef]

G. M. Morris, D. A. Zweig, “White-light Fourier transformations,” in Optical Signal Processing, J. L. Horner, ed. (Academic, New York, 1987).

Rhodes, W. T.

W. T. Rhodes, A. A. Sawchuk, “Incoherent optical processing,” in Optical Information Processing, S. H. Lee, ed. (Springer-Verlag, Berlin, 1981).
[CrossRef]

Sawchuk, A. A.

W. T. Rhodes, A. A. Sawchuk, “Incoherent optical processing,” in Optical Information Processing, S. H. Lee, ed. (Springer-Verlag, Berlin, 1981).
[CrossRef]

Zweig, D. A.

G. M. Morris, D. A. Zweig, “White-light Fourier transformations,” in Optical Signal Processing, J. L. Horner, ed. (Academic, New York, 1987).

Appl. Opt. (4)

Opt. Commun. (3)

R. Ferriere, J. P. Goedgebuer, “A spatially coherent achromatic Fourier transformer,” Opt. Commun. 42, 223–225 (1982).
[CrossRef]

C. Brophy, “Design of an all-glass achromatic Fourier transform lens,” Opt. Commun. 47, 364–368 (1983).
[CrossRef]

G. M. Morris, “An ideal achromatic Fourier processor,” Opt. Commun. 39, 143–147 (1981).
[CrossRef]

Other (3)

H. Bartelt, S. K. Case, R. Hauck, “Incoherent optical processing,” in Applications of Optical Fourier Transforms, H. Stark, ed. (Academic, New York, 1982).

W. T. Rhodes, A. A. Sawchuk, “Incoherent optical processing,” in Optical Information Processing, S. H. Lee, ed. (Springer-Verlag, Berlin, 1981).
[CrossRef]

G. M. Morris, D. A. Zweig, “White-light Fourier transformations,” in Optical Signal Processing, J. L. Horner, ed. (Academic, New York, 1987).

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Figures (6)

Fig. 1
Fig. 1

Imaging properties of a zone plate using broad-spectrum light: (a) the chromatic images of O formed by ZP1 are displayed in a volume V; and (b) the chromatic objects O(σ) forming V can be imaged through a zone plane ZP2 in a single achromatic image O′.

Fig. 2
Fig. 2

Setup for the achromatic Fourier transformer. The symbols used in the text are also shown. Note that the volume V″ lies in the object space of ZP2

Fig. 3
Fig. 3

Scale and axial position of the Fourier transform vary with σ in such a way that an arbitrary frequency always subtends the same angle from the optical center The hatched region of ZP2 represents the image of the volume V″ in Fig. 2 through ZP2 when a conventional but positive value of a is assumed.

Fig. 4
Fig. 4

Residual chromatic aberration of the system design in Fig. 2 for α = 1 (e.g., f = d2′ = Z0 = 300 mm) and with the following values for the parameter σ0: solid curve, σ0 = 1.94 μm−1; dashed curve, σ0 = 1.87 μm−1.

Fig. 5
Fig. 5

Same as in Fig. 3 but with σ 0 = σ 1 σ 2.

Fig. 6
Fig. 6

Plot of the residual chromatic aberration of the optical Fourier transformer in Fig. 2. Short-dashed curve: f = 150 mm, d2′ = 50 mm, and Z0 = −100 mm (i.e., α = −4.5 and D = 600 mm); dashed curve: f = 150 mm, d2′ = 45 mm, and Z0 = 65 mm (i.e., α = 7.7 and D = 975 mm); solid curve: f = 180 mm, d2′ = 50 mm, and Z0 = −80 mm (i.e., α = −8.1 and D = 898 mm).

Equations (31)

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( 1 / d 1 ) + ( 1 / d 1 ) = 1 / Z ,
d 1 ( σ ) = d 1 Z 0 σ / ( d 1 σ 0 - Z 0 σ ) .
d 2 ( σ ) = - d 2 Z 0 σ / ( d 2 σ 0 + Z 0 σ ) .
d ˙ 2 ( σ ) = - d 2 2 Z 0 σ 0 / ( d 2 σ 0 + Z 0 σ ) 2 ,
a 2 ( σ ) = - f 2 σ 0 / [ σ 0 ( l - 0 f ) - Z 0 σ ] ,
a ˙ 2 ( σ ) = - f 2 Z 0 σ 0 / [ σ 0 ( l - f ) - Z 0 σ ] 2 .
a 2 ( σ ) = d 2 ( σ ) .
a 2 ( σ 0 ) = d 2 ( σ 0 ) ,             a ˙ 2 ( σ 0 ) = d ˙ 2 ( σ 0 ) .
f 2 / ( l - f - Z 0 ) = d 2 Z 0 / ( d 2 + Z 0 ) , f 2 Z 0 / ( l - f - Z 0 ) 2 = d 2 2 Z 0 / ( d 2 + Z 0 ) 2 .
Z 0 = f 2 / Z 0 ,
l = 2 Z 0 + f + ( f 2 / d 2 ) .
U 0 ( x , y ) = t ( x , y ) exp [ - i π σ 0 ( x 2 + y 2 ) / Z 0 ] .
U 1 ( x , y ) = exp { i π σ ( x 2 + y 2 ) [ ( 1 / f ) - ( l / f 2 ) - σ 0 / σ Z 0 ) ] } × - t ( x , y ) exp [ - i π σ 0 ( x 2 + y 2 ) / Z 0 ] × exp [ - i 2 π σ ( x x + y y ) / f ] d x d y .
U 2 ( x , y ) = exp [ i π σ ( x 2 + y 2 ) / B ] - t ( x , y ) × exp [ i π ( x 2 + y 2 ) ( σ - σ 0 ) 2 / Z 0 ( 2 σ - σ 0 ) ] × exp [ - i 2 π A ( x x + y y ) ] d x d y ,
A = f σ 2 / [ d 2 Z 0 ( σ 0 - 2 σ ) ] ,
B = d 2 / [ 1 - f 2 σ / d 2 Z 0 ( σ 0 - 2 σ ) ] .
x = - Z 0 d 2 u / σ 0 f ,             y = - Z 0 d 2 v / σ 0 f .
U 3 ( x , y ) = exp [ i π σ ( x 2 + y 2 ) / ( B + R ) ] × - t ( x σ / B A , y σ / B A ) exp [ i π ( σ - σ 0 ) 2 σ 2 × ( x 2 + y 2 ) / Z 0 ( 2 σ - σ 0 ) B 2 A 2 ] × exp [ i 2 π σ W 20 ( x 2 + y 2 ) ] × exp [ i 2 π σ ( x x + y y ) / ( B + R ) ] d x d y ,
R ( σ ) = { ( 1 / d 2 ) - [ Z 0 ( σ - σ 0 ) 2 / f 2 σ σ 0 ] } - 1 .
x ( σ ) = - Z 0 f d 2 σ u / [ f 2 σ σ 0 - Z 0 d 2 ( σ - σ 0 ) 2 ] , y ( σ ) = - Z 0 f d 2 σ v / [ f 2 σ σ 0 - Z 0 d 2 ( σ - σ 0 ) 2 ] .
LCA 0 ( σ ; σ 0 ) = 100 [ R ( σ ) - d 2 ] / d 2 .
TCA 0 ( σ ; σ 0 ) = 100 [ x ( σ ) - x ( σ 0 ) ] / x ( σ 0 ) = 100 [ y ( σ ) - y ( σ 0 ) ] / y ( σ 0 ) .
LCA 0 ( σ ; σ 0 ) = TCA 0 ( σ ; σ 0 ) = 100 { [ f 2 σ σ 0 / Z 0 d 2 ( σ - σ 0 ) 2 ] - 1 } - 1 .
CA 0 ( σ 1 ; σ 0 ) = CA 0 ( σ 2 ; σ 0 )
σ 0 = σ 1 σ 2 ,
R ( σ m ) = [ R ( σ 1 ) + d 2 ] / 2.
CA m ( σ ) = 100 [ R ( σ ) - R ( σ m ) ] / R ( σ m ) = 100 [ x ( σ ) - x ( σ m ) ] / x ( σ m ) ,
CA m ( σ ) = [ 2 CA 0 ( σ ; σ 1 σ 2 ) - CA 0 ( σ 1 ; σ 1 σ 2 ) ] / [ 2 + CA 0 ( σ 1 ; σ 1 σ 2 ) ] .
σ / ( σ - σ 0 ) 2 = 2 σ 1 / ( σ 1 - σ 0 ) 2 ,
CA m ( σ 1 ) = CA m ( σ 2 ) = CA m ( σ 1 σ 2 ) = ( 1 / 2 ) CA 0 ( σ 1 ; σ 1 σ 2 ) .
D = l + f + d 2 = 2 Z 0 + 2 f + d 2 + ( f 2 / d 2 ) .

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